In this paper, we propose an extension of trace ratio based Manifold learning methods to deal with multidimensional data sets. Based on recent progress on the tensor-tensor product, we present a generalization of the trace ratio criterion by using the properties of the t-product. This will conduct us to introduce some new concepts such as Laplacian tensor and we will study formally the trace ratio problem by discuting the conditions for the exitence of solutions and optimality. Next, we will present a tensor Newton QR decomposition algorithm for solving the trace ratio problem. Manifold learning methods such as Laplacian eigenmaps, linear discriminant analysis and locally linear embedding will be formulated in a tensor representation and optimized by the proposed algorithm. Lastly, we will evaluate the performance of the different studied dimension reduction methods on several synthetic and real world data sets.

翻译：本文提出了一种基于迹比流形学习方法的扩展，用于处理多维数据集。基于张量-张量积的最新进展，我们利用t-积的性质给出了迹比准则的推广形式。这促使我们引入拉普拉斯张量等新概念，并通过讨论解的存在性与最优性条件，对迹比问题进行了形式化研究。随后，我们提出了一种用于求解迹比问题的张量牛顿QR分解算法。我们将以张量表示形式构建拉普拉斯特征映射、线性判别分析和局部线性嵌入等流形学习方法，并通过所提算法进行优化。最后，我们在多个合成数据集和真实世界数据集上评估了不同降维方法的性能。